A rolling body accelerates more slowly than a sliding one, because part of the energy goes into the rotation. Calculate the rolling acceleration from the slope angle and the inertia ratio. Notably the result depends neither on mass nor on radius but only on shape – which is why every solid sphere rolls equally fast, large or small.
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A rolling body accelerates more slowly than a sliding one, because part of the energy goes into the rotation. Calculate the rolling acceleration from the slope angle and the inertia ratio. Notably the result depends neither on mass nor on radius but only on shape – which is why every solid sphere rolls equally fast, large or small.
A solid cylinder rolls without slip down a plane inclined at α = 30°. Its inertia ratio is JS/(m·R²) = ½·m·R²/(m·R²) = 0.5, giving a = 9.80665 m/s² · sin 30°/(1 + 0.5) = 4.903 m/s²/1.5 = 3.269 m/s² – exactly two thirds of the acceleration of a frictionlessly sliding body. The same ramp gives a solid sphere with i = 0.4 the value a = 4.903/1.4 = 3.502 m/s², five sevenths, and a thin-walled tube with i = 1.0 only a = 4.903/2 = 2.452 m/s², exactly half. Released simultaneously, the sphere reaches the bottom first, then the cylinder, then the tube – regardless of their masses and diameters.
Pure rolling without slip is assumed: the distance unwound at the circumference exactly matches the path of the centre, so translation and rotation are coupled through the rolling condition. This requires sufficient static friction; if the friction coefficient is too low or the plane too steep the body starts to slide and the formula no longer holds – the calculator does not check that condition, because it would need the static friction coefficient. A rigid body with a rotationally symmetric cross-section, a rigid plane, and a rolling radius that stays constant throughout the motion are further assumed. Rolling resistance from deformation of body and surface, air resistance, bearing friction for wheels on axles, start-up transients and impacts, and bodies with an off-centre centre of mass that roll unevenly are all excluded.
Release a sphere, a solid cylinder and a tube down a ramp at the same time and they arrive in exactly that order – regardless of their masses and diameters. This calculator determines the rolling acceleration and makes the reason visible.
In pure rolling without slip, translation and rotation are coupled: the distance unwound at the circumference exactly matches the path of the centre. This rolling condition reads x = R·φ and, differentiated, v = R·ω. The released potential energy therefore splits into two parts – the kinetic energy of translation and that of rotation. Only the first accelerates the centre of mass, which is why a rolling body is slower than a sliding one. The split is set by the dimensionless inertia ratio JS/(m·R²), giving a = g·sin α/(1 + JS/(m·R²)).
Two rucksacks of equal weight: one carries everything close to your back, the other has the weight hanging far out. The second is harder to get moving although it weighs the same. It is the same for a rolling body – a tube has all its mass at the outside and must therefore put more energy into rotation than a solid sphere, whose mass sits closer to the axis.
a = g · sin α / (1 + JS/(m·R²))
Rolling acceleration: a = g·sin α/(1 + JS/(m·R²))Rolling condition: x = R·φ, hence v = R·ωInertia ratio: i = JS/(m·R²)Solid cylinder (i = 0.5): a = (2/3)·g·sin αSolid sphere (i = 0.4): a = (5/7)·g·sin αThin-walled tube (i = 1): a = ½·g·sin αFrictionless sliding (i = 0): a = g·sin αFinal speed after a ramp length L: v = √(2·a·L)| Symbol / input | Meaning |
|---|---|
| Rolling acceleration a | The acceleration of the centre of mass along the inclined plane for rolling without slip. It is always smaller than g·sin α, i.e. smaller than for a frictionlessly sliding body, because part of the potential energy is converted into rotational energy. Since the acceleration is constant, the usual laws of uniformly accelerated motion apply to speed and distance: v = a·t and s = ½·a·t². |
| Slope angle α | The angle of the inclined plane against the horizontal. At α = 0 the body does not start rolling by itself; at α = 90° there is no contact surface left and the body falls freely – that limiting case is no longer physically meaningful. Source: the drawing, the slope ratio via α = arctan(rise/run), or an inclinometer reading. |
| Inertia ratio JS/(m·R²) | The dimensionless ratio of the mass moment of inertia JS about the centroidal axis, the mass m and the square of the rolling radius R. It describes the shape of the body alone and is independent of its size. Typical values: solid cylinder or disc 0.5, solid sphere 0.4, hollow sphere 2/3 ≈ 0.667, thin-walled tube or ring 1.0, spoked wheel with a light hub close to 1.0. A body with i = 0 would be a frictionlessly sliding point mass with no rotational inertia. The larger i, the more energy sits in the rotation and the slower the body becomes. |
| Gravitational acceleration g | The local gravitational acceleration; nearly the same everywhere on Earth. Typical value 9.81 m/s², standard value 9.80665 m/s². Adjust only for very precise calculations or for other celestial bodies. |
α is the slope angle of the plane against the horizontal. The inertia ratio JS/(m·R²) describes the shape of the body alone: solid cylinder and disc 0.5, solid sphere 0.4, hollow sphere about 0.667, thin-walled tube or ring 1.0, spoked wheel with a light hub close to 1.0. g is the gravitational acceleration, typically 9.81 m/s². Mass and radius are not needed – they cancel out.
First assign the body to a basic shape and read off the matching inertia ratio; for assembled wheels, compute JS from the individual parts and then divide by m·R², where R is the rolling radius at the contact point and not, say, the hub diameter. Then enter the slope angle. Since the acceleration is constant, speed and distance follow directly from v = a·t and s = ½·a·t², and for a ramp length L the final speed is v = √(2·a·L). Finally, check whether the static friction is sufficient for rolling without slip at all – the calculator assumes it but does not verify it.
A solid cylinder rolls down a plane inclined at α = 30°. Its inertia ratio is 0.5, since JS = ½·m·R² divided by m·R² gives exactly ½. This yields a = 9.80665·sin 30°/(1+0.5) = 4.903/1.5 = 3.269 m/s², exactly two thirds of g·sin α. On the same ramp a solid sphere with i = 0.4 reaches 4.903/1.4 = 3.502 m/s², five sevenths, and a thin-walled tube with i = 1.0 only 4.903/2 = 2.452 m/s², exactly half. Over a 2 m ramp that gives final speeds of √(2·3.269·2) = 3.62 m/s for the cylinder and 3.13 m/s for the tube – a clearly measurable difference.
The most striking feature is what does not appear: neither mass nor radius. A marble and a bowling ball roll down the same ramp at exactly the same rate, because both quantities cancel out of the equation. What matters is solely the distribution of mass relative to the axis, expressed in the inertia ratio. The three limiting values are worth remembering: two thirds for the solid cylinder, five sevenths for the solid sphere, one half for the thin-walled tube. And a body with i = 0 – a frictionlessly sliding point mass – reaches the maximum g·sin α, which no rolling body can ever attain.
The calculation runs in metres, seconds and radians internally. Accelerations may be entered in m/s² or ft/s² and the angle in degrees or radians. The inertia ratio is dimensionless and entered as a decimal – 0.5, not 50 %. Note when forming the ratio in hand calculations: JS in kg·m², m in kg and R in metres, otherwise the result is not dimensionless.
Gravity chutes and roller conveyors for pipes, drums and rings; estimating run-out distances and final speeds on inclined transport sections; brakes and stops for rolling loads; experimental setups and demonstrations of the energy principle; designing wheel sets where light rims and heavy hubs improve acceleration; estimating how fast a part that has rolled away will become.
Pure rolling without slip is assumed, which requires sufficient static friction. If the friction coefficient is too low or the plane too steep the body starts to slide and the formula no longer holds; the calculator does not check that condition, because it would need the static friction coefficient. A rigid body with a rotationally symmetric cross-section, a rigid plane, and a rolling radius constant throughout the motion are further assumed. Rolling resistance from deformation of body and surface, air resistance, bearing friction for wheels on axles, start-up and impact transients, and bodies with an off-centre centre of mass that roll unevenly are not covered.
Common mistake: The most common misconception is expecting a heavier or larger body to roll faster – mass and radius cancel out completely. Second, people often compute with g·sin α, the value for frictionless sliding; that overestimates the rolling acceleration by 50 per cent for a solid cylinder and by 100 per cent for a thin-walled tube. Third, for an assembled wheel the hub diameter is used as R instead of the rolling radius at the contact point. And fourth, rolling without slip is assumed without checking the static friction – on steep ramps or smooth surfaces the body slides, and then a different calculation applies.
Gravity chutes and roller conveyors for pipes, drums and rings; estimating run-out distances and final speeds on inclined transport sections; brakes and stops for rolling loads; experimental setups and demonstrations of the energy principle; designing wheel sets where light rims and heavy hubs improve acceleration; estimating how fast a part that has rolled away will become.
α is the slope angle of the plane against the horizontal. The inertia ratio JS/(m·R²) describes the shape of the body alone: solid cylinder and disc 0.5, solid sphere 0.4, hollow sphere about 0.667, thin-walled tube or ring 1.0, spoked wheel with a light hub close to 1.0. g is the gravitational acceleration, typically 9.81 m/s². Mass and radius are not needed – they cancel out.
Pure rolling without slip is assumed, which requires sufficient static friction. If the friction coefficient is too low or the plane too steep the body starts to slide and the formula no longer holds; the calculator does not check that condition, because it would need the static friction coefficient. A rigid body with a rotationally symmetric cross-section, a rigid plane, and a rolling radius constant throughout the motion are further assumed. Rolling resistance from deformation of body and surface, air resistance, bearing friction for wheels on axles, start-up and impact transients, and bodies with an off-centre centre of mass that roll unevenly are not covered.